$\,15.\,$ Evaluate the following limits : $\,(i)\,\lim_{x \to 0}\frac{\sin 2x}{2x^2+x}\\=\lim_{x \to 0}\frac{\sin2x}{…

$\,12.\,$ Show : $\,(i)\,\lim_{x  \to 2}\,\frac{x^9-512}{x^4-16}=72$ Sol. $\,\,\lim_{x  \to 2}\,\frac{x^9-512}{x^4-16…

$\,11.\,$ Evaluate the following limits : $\,(i)\,\,\lim_{x \to 0}~~\frac{\sqrt{1+x}-\sqrt{1-x}}{x}$ Sol. $\,\,\lim_{…

$\,10.\,$ Evaluate the following limits : $\,(i)\,~~\lim_{x \to 2} \frac{x(x^2-4)}{x^2-5x+6}$  Sol. $\,~~~\lim_{x \to …

$\,6.\,$ Prove each of the following limits by intuitive and graphical methods : $\,(i)\, \lim_{x \to -2} ~4=4.$ Sol.…

$\,1.\,$ Give the definition of limit of a function. Sol. A limit of a function $\,f(x)\,$ can be defined as a value,…

$\,14.\,$ Evaluate : $\,(i)\,\lim_{x  \to 0}\frac{\sin x}{ \log_e(1+x)^{\frac 12}}\\=\lim_{x  \to 0}\frac{\sin x}{\fr…

$\,13.\,$ Prove : $\,(i)\,\lim_{x  \to 0}\frac{\log(1+2x)}{\sin3x}=\frac 23$ Sol. $\,\lim_{x  \to 0}\frac{\log(1+2x)}…

$\,(xi)\lim_{x  \to a}\,\frac{\sin x-\sin a}{x-a}=\cos a$ Sol. $\,\lim_{x  \to a}\,\frac{\sin x-\sin a}{x-a}\\=\lim_{…

$\,11.\,$ Prove : $\,(i)\,\, \lim_{x \to 0}\frac{\sin \alpha x^{\circ}}{\sin\beta x^{\circ}}=\frac{\alpha}{\beta}$ So…

$\,10(ix)\,\lim_{h  \to 0}\frac{\sin(x+h)-\sin x}{h}$ Sol. $\,\,~~\lim_{h  \to 0}\frac{\sin(x+h)-\sin x}{h}\\=\lim_{h …

$\,10.\,$ Evaluate the following limits : $\,(i)\,\lim_{x  \to 0}\frac{\sqrt{\cos x}-\sqrt[3]{\cos x}}{\sin^2x}$ Sol. …

$\,6.\,$ Evaluate : $\,\lim_{x \to 0} \frac{\sqrt{1+x^2}-\sqrt{1+x}}{\sqrt{1+x^3}-\sqrt{1+x}}$ Sol. $\,\lim_{x \to 0} …